guler generalized lodf

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Abstract Distribution factors play a key role in many system

security analysis and market applications. The injection shift

factors (ISFs) are the basic factors that serve as building blocks

of the other distribution factors. The line outage distribution

factors (LODFs) may be computed using the ISFs and, in fact,

may be iteratively evaluated when more than one line outage is

considered. The prominent role of cascading outages in recent

black-outs has created a need in security applications for

evaluating LODFs under multiple-line outages. In this letter, we

present an analytic, closed-form expression for and the

computationally efficient evaluation ofLODFs under multiple-

line outages.

Index Termspower transfer distribution factors, line outage

distribution factors, multiple-line outages, system security.

I. INTRODUCTIONDistribution factors are linear approximations of the

sensitivities of specific system variables with respect to

changes in nodal injections and withdrawals [1]-[5]. While the

line outage distribution factors (LODFs) are well understood

[1], the evaluation ofLODFs under multiple-line outages has

received little attention. Given the usefulness ofLODFsin the

study of security with many outaged lines, such as in blackouts

impacting large geographic regions, we focus on the fast

evaluation of LODFs under multiple-line outages the

generalized LODFs or GLODFs. This letter presents an

analytic, closed-form expression for, and the computationally

efficient evaluation of, GLODFs.

II. BASIC DISTRIBUTION FACTORSWe consider a power system consisting of (N+1) buses and

L lines. We denote by { ,1, ,0 N= KN the set of buses, with

the bus 0 being the slack bus, and by { }1,.., L= l lL the set of

transmission lines. We associate with each linem

l L , the

ordered pair of nodes (im, jm). We us the convention that the

direction of the real power flowm

fl on the line ml is from mi

tom

j .

The ISFk

i l of line kl is the (approximate) sensitivity of thechange in the line

kl real power flow

kfl with respect to a

change in the injection pi at some node iN and the

withdrawal of an equal change amount at the slack bus. Under

the lossless conditions and the typical assumptions used in DC

power flow, we construct the ISFmatrix 1dB A B@ [5], with

Manuscript received June 26, 2006. Research is supported in part by

PSERC.T. Guler and G. Gross are with the Department of Electrical and

Computer Engineering, University of Illinois at Urbana-Champaign,

Urbana, IL 61801 USA (e-mail:[email protected], [email protected]). M. Liu

is with CRAI, Boston, MA 02116 USA (e-mail: [email protected])

with L LdB being the branch susceptance matrix,

L NA the reduced incidence matrix and N NB the

reduced nodal susceptance matrix.

We evaluate the power transfer distribution factors (PTDFs)

by introducing notation for transactions. The impact of a ? t-

MW transaction from node i to nodej, denoted by the ordered

triplet { }, , ,w i j t @ onk

fl is kw

f l , and is determined by

k k

w wf t = l l ,

where thePTDFk

w l is defined as [5]

k k k

w i j l l l@ . (2)

For the linem

l outage, we evaluate the impact ( )mk

fl

l on

the flowk

fl on line kl using the LODF( )m

k

l

l which specifies

the fraction of the pre-outage real power flow on the linem

l

redistributed to the linek

l [5] and is given by:

( )

( ) ( )

( )( )1

m m

k km

k m

m m

w

w

f

f

=

l ll ll

l ll l

@ ,k m

l l . (3)

Here, ( ) , ,m m mw i j t = l denotes the transactionbetween the

terminal nodes ofk

l . As long as ( ) 1mm

w

l

l ,( )m

k

l

l is defined.

The linem

l outage results in a topology change and

necessitates reevaluation of the post-outage networkPTDFs.

We use the notation ( )( )ml

to denote the value of the vari-

able with the linem

l outaged, as in (3). The pre- and post-

outage PTDFs,k

wl and ( )( )m

k

wl

l , respectively, are related by [5]

( )( ) ( )m m

k k k m

w w w +l l

l l l l@ . (4)

We use the distribution factors introduced in this section to

generalize theLODFexpression for multiple-line outages.

III. DERIVATION OF GLODFSWe first revisit the single-line outage case and examine how

the outage impacts may be simulated by net injection and

withdrawal changes. The line ( )1 1 1,i j=% % %l outage changes the

real power flow in the post-outage network on each line

connected to 1i% by the fraction of1

f%l . We simulate this impact

by introducing ( ) ( ){1 11 1, ,w i j t = % %% %l l in the pre-outage

network. The injection ( )1t %l adds a change( )

( )11

1

wt

%l

%l%l on

the line 1%l flow and a net flow change of ( )( ) ( )1

111 w t

%l%l %l on

all the other lines but 1%l that are connected to node 1i% . By

selecting ( )1t %l to satisfy( )( ) ( )11 1

11w

t f =%l

% %l l%l , (5)

the transaction ( )1w %l changes the flow kfl , 1k %l l , by

( ) ( ) ( ) ( ) ( )( )1 1 1 11 1

1

1

1k k k

w w w wf t f

= =

% % % %l l l l

% %l l l l l%l . (6)

Generalized Line Outage Distribution Factors

Teoman Gler, Student Member, IEEE, and George Gross,

Fellow, IEEEand Minghai Liu,Member, IEEE


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In terms of (3), the bracketed term in (6) is( )1

k

%l

l , and so ( )1w %l

with ( )1t %l given by (6) simulates the line 1%l outage impacts.We proceed with the generalization for multiple-line outages

by next considering the case of the outages of the two

lines 1%l and 2

%l .We simulate the impacts on ,k

fl by introducing

( )1w %l and ( )2w %l , taking explicitly into account the inter-

actions between these two transactions in specifying ( )1t %l

and ( )2t %l , as shown in Fig. 1. We set ( )1t %l to satisfy

( )( )( )

( ) ( ) ( )( )2

1 2

1 11 .1

wt f =

%l% %l l% %l l

%l (7)

Analogously, we select ( )2t %l to satisfy

( )( )( )

( ) ( ) ( )( )1

2 1

2 22 .1

wt f =

%l% %l l% %l l

%l (8)

We rewrite (7) and (8) using the relations in (3) and (4) as

( ) ( )

( ) ( )

( )( )

1 2

11 1

1 2

2

2 2

1

2 .

w w

w w

ft

ft

=

I

% %l l%% % ll l

% %l l%l

% %l l

%l

%l(9)

As long as the matrix in (9) is nonsingular, we determine

( )1t %l and ( )2t %l by solving the linear system.

Fig. 1. The impacts of the transactions ( )1

w %l and ( )2w%l .

In the inductive process to generalize the result for the case

of multiple-line outages, we assume that the impacts of a set ofa1 outaged lines ( ) }1 11 , , % % %l L lL = are simulated with

a1 transactions whose amounts are specified by

( ) ( ) ( )1 1 1 =

I t f% L , (10)

where, ( ) ( ) ( )111 , ,T

t t

= t%% K ll , ( ) 1 11

, ,T

f f

= f % %l lK

and( )

( ) ( )

( ) ( )

1 1

1 1

1

1 1

1 1

w w

w w

=

% %l l

% %l l

%% %l l

% %l l

L

M O M

L

L

, with( )1

I %L nonsingular.

We now consider the additional line ( )1 % %l L outage. The set

of outaged lines is ( ) ( ) { }1 % % %U lL L= . Reasoning along thelines used in the two-line outage analysis, ( )1 t is given by

( )( )

( )( ) ( )( )

( )1 1 1 .

= I t f

% %l l%

L (11)

We capture the impacts of the outages of the ( )1 %L elements

on %l by using the analogue of (8) and determine ( )t %l from

( )( )( )( )( ) ( ) ( ) ( )( )1 1 .1 w ft

=

% %%l%l%l

%lL L

(12)

The superscript( )( )1 %L denotes the network with the

elements of ( )1 %L outaged. We define

( ) ( )

1 1

,..,Tw w

b% %l l

% %l l@

and( ) ( )1 1

,..,w w

T

c% %l l

% %l l@ and rewrite (11) and (12) as

( ) ( )

( )( ) ( )( ) ( )( )( ) ( ) ( ) ( ) ( )( )

1

1

1

1 1 1

1

1

1

1 ,

w

w

f

t t f

+ =

+ =

T

T

I t b c t f

c I f b

%l% % %l l

%l%% %l l

% %l l

L

L

(13)

which may be simplified to

( ) ( ) ( ) =

I t f% L ,( )

( )

( )1

.w

T

b

c

%

%% l

%l

@

L

L(14)

So long as( )

I %L is nonsingular, we use (14) to solve for

( )t and so simulate the impacts of the line outages.

This development for specifying the appropriate values of

the transactions is used to provide the GLODFexpression. For

any line ( )k %l L , we define ( )

( )k

%

l

L

, whose elements are the

GLODFs with the lines in ( )%L outaged, with the interactions

between the outaged lines fully considered. The change in the

real power flow of linekl is

( )( )( ) ( )( )

( )k k

T

f

f% %

l l@L L

, ( )k %l L (15)

However, the combined impacts on linek

l of the a

transactions with the ( )t specified by (14) is

( )( )( ) ( ) ( )

( )1 , ,

k k k

w wf

= t

% % %l l

l l lK

L

. (16)

Therefore, for( )

I %L nonsingular, we rewrite (16) as

( )( )( ) ( ) ( )

( ) ( )1

1

, ,k k k

w wf

= I f

%% %l l

%l l l

LL

L . (17)

It follows from (15) that

( )( )k

%

l

L

is the solution of

( )

( )( ) ( ) ( )1, ,

k k k

TT w w

= I

% % %l l%

l l lLL

L , (18)

and is defined whenever( )

I %L is nonsingular. The case

for singular( )

I %L indicates that the outage of the

lines in ( )%L separates the system into two or more islands.

The analysis of such cases is treated in [6]. In fact, a simple

case is the outage of a line whose PTDFequals one. In this

case, the outage of the line results in the creation of two

separate subsystems. When the outaged line whose PTDFis

unity happens to be a radial tie, the outage results in the

isolation of the radial node.The relation (18) provides an analytic, closed-form

expression for the GLODFs. Since the GLODFis expressed in

terms of the pre-outage network parameters, we avoid the need

to evaluate the post-outage network parameters. A key

advantage in the deployment of GLODFs is the ability to

evaluate the post-outage flows on specific lines of interest

without the need to determine the post-outage network states.

The proposed LODFextension permits the GLODFevaluation

through a computationally efficient procedure which involves

the solution of a system of linear equations whose dimension is

2%

( ) ( )1 22 2 2

w wf f f+ +

% %l l

% % %l l l

( ) ( )1 21 1 1

w wf f f+ +

% %l l

% % %l l l

( ) ( )1 2 k k k

w wf f f+ +

% %l l

l l l

k k

1% 2%

( )2t %l

( )2t %l 1

j%

( )1t %l ( )1t %l


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is the number of line outages.

IV. SUMMARYThe prominent role of cascading outages in recent blackouts

has created a critical need in security applications for the rapid

assessment of multiple-line outage impacts. We developed a

closed-form analytic expression forGLODFs under multiple-line

outages without the reevaluation of post-outage network

system parameters. This general expression allows the

computationally efficient evaluation of GLODFs for security

application purposes. A very useful application of GLODFs is

in the detection of island formation and the identification of

causal factors under multiple-line outages [6].

REFERENCES

[1] A. Wood and B. Wollenberg, Power Generation Operation andControl, 2nd edition, New York: John Wiley & Sons, p.422, 1996.

[2] R. Baldick, Variation of distribution factors with loading, IEEETrans. on Power Syst., vol. 18, pp. 13161323, Nov. 2003.

[3] F.D. Galiana, Bound estimates of the severity of line outages,IEEE Trans. on Power Apparatus and Systems, vol. PAS-103, pp.

26122624, Feb. 1984.[4] M. Liu and G. Gross, Effectiveness of the distribution factor

approximations used in congestion modeling, InProceedings of the

14th Power Systems Computation Conference, Seville, 24-28 June,

2002.

[5] M. Liu and G. Gross, Role of distribution factors in congestionrevenue rights applications, IEEE Trans. on Power Syst., vol. 19,

pp. 802-810, May 2004.

[6] T. Guler and G. Gross, Detection of island formation andidentification of causal factors under multiple line outages, to be

published in IEEE Trans. on Power Syst.

guler generalized lodf

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